How are US students performing compared to other nations?...

To put it simply, not good. Not good at all. And in fact, getting worse.
A recent article in The Economist explains:
 
It is easy to be cynical about school-test results, particularly when you are grading the performance of something as complex as a country's education system. Undaunted, every three years the Programme for International Sudent Assessment (PISA), which is run by the OECD club of mainly rich countries, tests more than half a million 15-year-olds in three subjects- math, readin and science- to giva a snapshot of national school policies. The latest results were published on December 6th and again show stellar achievement in East Asia. Singaporean pupils are roughly three years ahead of American ones in math. Some argure that differences in national scores are a result of parenting and innate culture, and therefore that policy makers can do little to improe pupils' performance.

Culture matters, but it is not as if success in PISA is the preserve of East Asia. Estonia scores highly enough to beat the rest of Europe and achieves similar results to Japan.

PISA teaches what does not work. Spending more money, for example, is associated with higher scores, but only in poorer countries. Among those that already spend more than $50,000 per pupil throughout their time in school, money alone brings no improvement. Private schools are no exception, at least when it comes to PISA.
The exercise also tells you what does work, and its most important insight is that what matters most is what happens in the classroom. The successful children are those who are exposed to good teaching more often. Having pupils turn up is a start. In poor countries this often means expanding access for girls. In richer countries it means cutting dropout rates and truancy; Italian pupils do poorly partly because more than half of them skip school at least once a fortnight. Having teachers turn up also helps. One reason why Buenos Aires saw the biggest rise in PISA scores of any area is because the city curbed teachers’ strikes by offering them a deal: it would treat teachers as professionals if they behaved as such. The city improved training and pay. Teachers agreed that merit, not their unions, would determine promotion. Improving the quality of teaching is harder. Who becomes a teacher makes a difference. Australia’s decline in PISA coincides with a fall in the exam results of teacher-training applicants. And what teachers learn about the job is at least as important. Evidence-based methods of instruction, practice, coaching from experienced teachers and feedback are all part of making good teachers.

Poor students tend to do less well in PISA. But the effects of poverty can be overcome. The influence of family background on test scores fell by more in America than in any other OECD country over the past decade. This partly reflects the growth of excellent autonomous but publicly funded charter schools in big cities. Successive presidents’ efforts to hold schools accountable have had some impact, too. In Estonia nearly half the poorest children achieve results that would place them in the top quarter across the OECD. A reason for this is a lack of selection by ability. Many of the top-performing school systems delay the start of formal education until the age of six or seven, focusing instead on play-based education. But they then make students learn academic subjects until about 16. Even in Singapore, where pupils can opt earlier for a vocational track, schools insist on academic rigour as well as practical work.

​Concentrate there at the back
Like spoilt children who have failed an exam, some policymakers claim the PISA tests are unfair. Certainly, PISA does not capture all of what matters in education. It offers clues rather than guarantees of what works. It is the fair, rigorous and useful work of technocrats. Yet politicians who ignore it are turning their back on powerful truths.